Optimization in graphical small cancellation theory

Abstract: Gromov (2003) constructed finitely generated groups whose Cayley graphs contain all graphs from a given infinite sequence of expander graphs of unbounded girth and bounded diameter-to-girth ratio. These so-called Gromov monster groups provide examples of finitely generated groups that do not coarsely embed into Hilbert space, among other interesting properties. If graphs in Gromov's construction admit graphical small cancellation labellings, then one gets similar examples of Cayley graphs containing all the graphs of the family as isometric subgraphs. Osajda (2020) recently showed how to obtain such labellings using the probabilistic method. In this short note, we simplify Osajda's approach, decreasing the number of generators of the resulting group significantly.